Some known air compressors operate using an internal combustion engine as its motive-power source. Such air compressors comprise, in addition to the engine, an air-compressing part, such as a piston that is reciprocally slidable inside a cylinder, an air-storage tank, which stores compressed air discharged from the cylinder, and a pressure-detecting means, which detects the air pressure inside the air-storage tank. Such air compressors are configured such that, at the point in time when the air pressure inside the air-storage tank has increased to a set pressure, the engine is switched to an idling operation. For example, Japanese Laid-open Patent Publication 2003-239863 discloses a method for operating such an air compressor, in which compressed air generated during an idling operation is externally discharged and serves as a cooling air draft to cool a high-temperature component of the air compressor.